Matakuliah
Tahun
: Matrix Algebra for Statistics
: 2009
Invers Matriks Positif dan Non-Positif
Pertemuan 12
Introduction
• A nonnegative matrix is a matrix in
which all the elements are equal to or
greater than zero
• A positive matrix is a matrix in which
all the elements are greater than zero
• M-matrix is a Z-matrix with
eigenvalues whose real parts are
positive
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• A rectangular non-negative matrix can be
approximated by a decomposition with two
other non-negative matrix via non-negative
matrix factorization
• A matrix that is both non-negative and
positive semidefinite is called a doubly
non-negative matrix
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Z-matrices
• The Z-matrices are those matrices
whose off-diagonal entries are less
than or equal to zero; that is,
a Z-matrix Z satisfies
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Inverse
• An inverse of a non-singular so-called Mmatrix is a non-negative matrix
• If the non-singular M-matrix is also
symmetric then it is called a Stieltjes
matrix
• The inverse of a non-negative matrix is
usually not non-negative. An exception is
the non-negative monomial matrices
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Stieltjes matrix and monomial matrix
• A Stieltjes matrix is a real symmetric
positive definite matrix with nonpositve offdiagonal entries. A Stieltjes matrix is
necessarily an M-marix
• monomial matrix is a matrix with the
same nonzero pattern as a permutation
matrix, i.e. there is exactly one nonzero
entry in each row and each column.
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The Banachiewicz Identity and Inverse
Positive Matrices
• An (n−1)×(n−1) real matrix E is given with
n ≥2. Assume that E is inverse-positive,
i.e., E−1 > 0. Now to this E , one row and
one columnare attached to the bottom and
to the right end, respectively. Let
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where F is in R(n−1)×1, G in R1×(n−1), with ann
and being scalars. Then the Banachiewicz
identity gives
Where
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Example
and ann=1
E is not a Z-matrix and
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E is inverse-positive
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